Geography of the level sets of the Brownian sheet (Q1326354)

We describe geometric properties of \(\{W>\alpha\}\), where \(W\) is a standard real-valued Brownian sheet, in the neighborhood of the first hit \(P\) of the level set \(\{W>\alpha\}\) along a straight line or smooth monotone curve \(L\). In such a neighborhood we use a decomposition of the form \(W(s,t)=\alpha-b(s)+B(t)+x(s,t)\), where \(b(s)\) and \(B(t)\) are particular diffusion processes and \(x(s,t)\) is comparatively small, to show that \(P\) is not on the boundary of any connected component of \(\{W>\alpha\}\). Rather, components of this set form clusters near \(P\). An integral test for thorn-shaped neighborhoods of \(L\) with tip at \(P\) that do not meet \(\{W>\alpha\}\) is given. We then analyse the position and size of clusters and individual connected components of \(\{W>\alpha\}\) near such a thorn, giving upper bounds on their height, width and the space between clusters. This provides a local picture of the level set. Our calculations are based on estimates of the length of excursions of \(B\) and \(b\) and an accounting of the error term \(x\).